Читать книгу Vibroacoustic Simulation - Alexander Peiffer - Страница 33

A similar procedure is applied to the springs and the related stiffness matrix. Every spring k_i connects different points in space, so there are at least two nodes involved, unless the spring is connected to a rigid wall. Inspecting the spring in Figure 1.18 the force balance is:

(1.95)

Figure 1.18 Spring and damper connection of two nodes. Source: Alexander Peiffer.

Springs connected to rigid walls are treated differently. In that case the DOF at the wall is constrained. Here for example u1=0, and we get for Equation (1.95)

(1.96)

If the spring is connected to the wall, only u₂ changes with applied forces, and the reaction force in the constraint is −ksu2. Practically that is the reaction force that must be provided by the wall to keep the node in place. In the finite element terminology these constraints are called single point constraints (SPC).

The local matrices must be distributed among the global stiffness matrix [K]. For better visibility this is done for springs 2 and 4. The place of the local matrix of spring 2 in the global stiffness matrix follows from the u₁ and u₂, the position of spring 4 from v₁ and v₃.

(1.97)

For the system from Figure 1.17 and putting all free and constraint springs together we get:

(1.98)

With this procedure the matrix formulation of the equation of motion can be created. A similar approach can be used if other elements like dampers are involved. Generally the local elements can be everything that can be expressed by a dynamic stiffness matrix [D(ω)] and can be added into a global matrix, independent from the fact if it comes from other models, simulation or test.